M 112 Short Course in Calculus

Chapter 1 – Functions and ChangeSections 1.5 Exponential FunctionsV. J. Motto

1.4 Exponential Functions

An exponential function is a function of the form

Where a ≠ 0, b > 0, and b ≠ 1. The exponent must be a variable.

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xy = a*b

Illustration 1

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Illustration 2: Different b’s, b > 0

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What conclusions can we make looking at these graphs?Use your calculator to sketch these graphs!

Illustration 3: Different b’s, 0 < b <1

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Graph these on your calculator. What can we conclude? Are there other ways to write these equations?

Comments on y = bx

All exponential graphs Go through the point (0, 1)Go through the point (1, b)Are asymptotic to x-axis.

The graph f(x) = b-x =

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1 x

b

Illustration 4: (page 39)

Population of Nevada 2000-2006 Dividing each year’s population by the previous year’s population gives us

We find a common ratio!

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Illustration 4 (continued)

These functions are called exponential growth functions. As t increases P rapidly increasing.

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Thus, the modeling equation is P(t) = 2.020(1.036)t

Illustration 5: Drugs in the Body

Suppose Q = f(t), where Q is the quantity of ampicillin, in mg, in the bloodstream at time t hours since the drug was given. At t = 0, we have Q = 250. Since the quantity remaining at the end of each hour is 60% of the quantity remaining the hour before we have

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Illustration 5: (continued)

You should observe that the values are decreasing! The function Q = f(t) = 250(0.6)t

Is an exponential decay function. As t increases, the function values get arbitrarily close to zero.

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Comments

The largest possible domain for the exponential function is all real numbers, provided a >0.

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Linear vs Exponential

Linear function has a constant rate of change

An exponential function has a constant percent, or relative, rate of change.

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Example 1: (page 41)

The amount of adrenaline in the body can change rapidly. Suppose the initial amount is 25 mg. Find a formula for A, the amount in mg, at time t minutes later if A is

a) Increasing by 0.4 mg per minuteb) Decreasing by 0.4 mg per minutec) Increasing by 3% per minuted) Decreasing by 3% per minute

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Example 1: (continued)

Solutiona)A = 25 + 0.4 t - linear increaseb)A = 25 – 0.4t - linear decreasec)A = 25(1.03)t - exponential growthd)A = 25(0.97)t - exponential decay

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Example 3: (page 42)

Which of the following table sof values could correspond to an exponential function or linear function? Find the function.

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Example 3: (continued)

a) f(x) = 15(1.5)x, (common ratio)b) g is not linear and g is not exponentialc) h(x) = 5.3 + 1.2x

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Research Homework

Search the internet (or mathematics books you own) and find a demonstration that discovers the value of e.

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